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CHARACTER AND OBJECT

Published online by Cambridge University Press:  20 June 2016

JEREMY AVIGAD  and
REBECCA MORRIS
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Abstract

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.

In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 3 , September 2016 , pp. 480 - 510
Copyright
Copyright © Association for Symbolic Logic 2016 

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